Question: Solve for $x$ : $ 2|x + 5| + 8 = -3|x + 5| + 10 $
Solution: Add $ {3|x + 5|} $ to both sides: $ \begin{eqnarray} 2|x + 5| + 8 &=& -3|x + 5| + 10 \\ \\ { + 3|x + 5|} && { + 3|x + 5|} \\ \\ 5|x + 5| + 8 &=& 10 \end{eqnarray} $ Subtract ${8}$ from both sides: $ \begin{eqnarray} 5|x + 5| + 8 &=& 10 \\ \\ { - 8} &=& { - 8} \\ \\ 5|x + 5| &=& 2 \end{eqnarray} $ Divide both sides by ${5}$ $ \dfrac{5|x + 5|} {{5}} = \dfrac{2} {{5}} $ Simplify: $ |x + 5| = \dfrac{2}{5}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 5 = -\dfrac{2}{5} $ or $ x + 5 = \dfrac{2}{5} $ Solve for the solution where $x + 5$ is negative: $ x + 5 = -\dfrac{2}{5} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& -\dfrac{2}{5} \\ \\ {- 5} && {- 5} \\ \\ x &=& -\dfrac{2}{5} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $5$ $ x = - \dfrac{2}{5} {- \dfrac{25}{5}} $ $ x = -\dfrac{27}{5} $ Then calculate the solution where $x + 5$ is positive: $ x + 5 = \dfrac{2}{5} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& \dfrac{2}{5} \\ \\ {- 5} && {- 5} \\ \\ x &=& \dfrac{2}{5} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $5$ $ x = \dfrac{2}{5} {- \dfrac{25}{5}} $ $ x = -\dfrac{23}{5} $ Thus, the correct answer is $x = -\dfrac{27}{5} $ or $x = -\dfrac{23}{5} $.